Let Af denote the form of Af.

Let denote the Bézier curve determined by the points P < sub > 0 </ sub >, P < sub > 1 </ sub >, ..., P < sub > n </ sub >.

Let X denote a Cauchy distributed random variable.

Let w denote the weight per unit length of the chain, then the weight of the chain has magnitude

Let denote the equivalence class to which a belongs.

Let '~' denote an equivalence relation over some nonempty set A, called the universe or underlying set.

Let G denote the set of bijective functions over A that preserve the partition structure of A: ∀ x ∈ A ∀ g ∈ G ( g ( x ) ∈ ).

Let G be a set and let "~" denote an equivalence relation over G. Then we can form a groupoid representing this equivalence relation as follows.

Let R denote the field of real numbers.

Let n denote a complete set of ( discrete ) quantum numbers for specifying single-particle states ( for example, for the particle in a box problem we can take n to be the quantized wave vector of the wavefunction.

Let ε ( n ) denote the energy of a particle in state n. As the particles do not interact, the total energy of the system is the sum of the single-particle energies.

Let denote the space of scoring functions.

Let the line of symmetry intersect the parabola at point Q, and denote the focus as point F and its distance from point Q as f. Let the perpendicular to the line of symmetry, through the focus, intersect the parabola at a point T. Then ( 1 ) the distance from F to T is 2f, and ( 2 ) a tangent to the parabola at point T intersects the line of symmetry at a 45 ° angle.

Let us denote the time at which it is decided that the compromise occurred as T.

Let denote the sequence of convergents to the continued fraction for.

Let us denote the mutually orthogonal single-particle states by and so on.

That is, Alice has one half, a, and Bob has the other half, b. Let c denote the qubit Alice wishes to transmit to Bob.

Let H be a Hilbert space, and let H * denote its dual space, consisting of all continuous linear functionals from H into the field R or C. If x is an element of H, then the function φ < sub > x </ sub >, defined by

If V is a real vector space, then we replace V by its complexification V ⊗< sub > R </ sub > C and let g denote the induced bilinear form on V ⊗< sub > R </ sub > C. Let W be a maximal isotropic subspace, i. e. a maximal subspace of V such that g |< sub > W </ sub > = 0.

Let Q ( x ) denote the number of square-free ( quadratfrei ) integers between 1 and x.

A possible definition of spoiling based on vote splitting is as follows: Let W denote the candidate who wins the election, and let X and S denote two other candidates.

Let π < sub > 2 </ sub >( x ) denote the number of primes p ≤ x such that p + 2 is also prime.

Let be a sequence of independent and identically distributed variables with distribution function F and let denote the maximum.

Next, take a Hamiltonian invariant under T. Let | a > and T | a > be two quantum states of the same energy.

Let V be a Hamiltonian representing a weak physical disturbance, such as a potential energy produced by an external field.

Let H ( 0 ) be the Hamiltonian completely restricted either in the low-energy subspace or in the high-energy subspace, such that there is no matrix element in H ( 0 ) connecting the low-and the high-energy subspaces, i. e. if.

Let us consider a square array of classical spins which may only take two positions: + 1 and − 1, at a certain temperature, interacting through the Ising classical Hamiltonian:

Let be an observable of a dynamical system with Hamiltonian subject to thermal fluctuations.

Let H < sup > i </ sup > be the Hamiltonian of the i-th such system.

Let denote a representative point in the phase space, and be its image under the Hamiltonian flow at time t. The time average of f is defined to be

Let me set the record this time, and let me get back OK, so the German will give me the exclusive.

Let evildoers contemplate their ways, and let every man beware ''!!

Let us not try to key them out at this stage of the game, and let us just call them Bombus.

Let N be a positive integer and let V be the space of all N times continuously differentiable functions F on the real line which satisfy the differential equation Af where Af are some fixed constants.

Let Q be a nonsingular quadric surface bearing reguli Af and Af, and let **zg be a Af curve of order K on Q.

Let her call Crosson if she wanted to, let Crosson raise the roof or even can him, he didn't care.

Let not your heart be troubled, neither let it be afraid ''.

Let us focus on an atom of calcium from the tip of the bone of my finger and let us suppose that I swallow a magic Alice In Wonderland growing pill.

Let her out, let her out -- that would be the solution, wouldn't it??

Let him call for the elders of the church, and let them pray over him, anointing him with oil in the name of the Lord ; and the prayer of faith will save the sick man, and the Lord will raise him up ; and if he has committed sins, he will be forgiven.

Let no man add to these, neither let him take out from these.

Let and let.

Let the function g ( t ) be the altitude of the car at time t, and let the function f ( h ) be the temperature h kilometers above sea level.

Let and be differentiable functions, and let D be the total derivative operator.

Let M be a smooth manifold and let x be a point in M. Let T < sub > x </ sub > M be the tangent space at x.

Let M be a smooth manifold and let x be a point in M. Let I < sub > x </ sub > be the ideal of all functions in C < sup >∞</ sup >( M ) vanishing at x, and let I < sub > x </ sub >< sup > 2 </ sup > be the set of functions of the form, where f < sub > i </ sub >, g < sub > i </ sub > ∈ I < sub > x </ sub >.

Let M be a smooth manifold and let f ∈ C < sup >∞</ sup >( M ) be a smooth function.

Let them be thousands, let them drown themselves in their own blood.

* Let H be a group, and let G be the direct product H × H. Then the subgroups

Let x, y, z be a system of Cartesian coordinates in 3-dimensional Euclidean space, and let i, j, k be the corresponding basis of unit vectors.

Let X be a nonempty set, and let.